Proof. Let SSS be the sum of (1), Sn=a1+a2+⋯+ansubscriptSnsubscripta1subscripta2normal-⋯subscriptanS_{n}=a_{1}\!+\!a_{2}\!+\cdots+\!a_{n} the nthsuperscriptnthn^{\mathrm{th}}partial sum of (1) and Rn+1=S-Sn=an+1+an+2+⋯subscriptRn1SsubscriptSnsubscriptan1subscriptan2normal-⋯R_{{n+1}}=S\!-\!S_{n}=a_{{n+1}}\!+\!a_{{n+2}}\!+\cdots the corresponding remainder term. Then we have

Then the series g1+g2+g3+⋯subscriptg1subscriptg2subscriptg3normal-⋯g_{1}\!+\!g_{2}\!+\!g_{3}\!+\cdots fulfils the requirements in the theorem. Its terms gnsubscriptgng_{n} are positive. Further, it converges because its nthsuperscriptnthn^{\mathrm{th}} partial sum is equal to
R1-Rn+1subscriptR1subscriptRn1\sqrt{R_{1}}\!-\!\sqrt{R_{{n+1}}} which tends to the limitR1=SsubscriptR1S\sqrt{R_{1}}=\sqrt{S} as n→∞normal-→nn\to\infty since Rn+1→0normal-→subscriptRn10R_{{n+1}}\to 0; this implies also (2).